First slide

Speed of wave

Question

A uniform rope of length $L$ and mass $m_{1}$ hangs vertically from a rigid support. A block of mass $m_{2}$ is attached to the free end of the rope. A transverse pulse of wavelength $λ_{1}$ is produced at the lower end of the rope. The wavelength of the pulse when it reaches the top of the rope is $λ_{2}$ . The ratio $\frac{λ_{2}}{λ_{1}}$ is

Moderate

A

$\sqrt{\frac{m_{2}}{m_{1}}}$
B

$\sqrt{\frac{m_{1} + m_{2}}{m_{1}}}$
C

$\sqrt{\frac{m_{1}}{m_{2}}}$
D

$\sqrt{\frac{m_{1} + m_{2}}{m_{2}}}$

Solution

$Wavelength of pulse at the lower$ $end (λ_{1}) \propto velocity (v_{1}) = \sqrt{\frac{T_{1}}{μ}}$
$Similarly, λ_{2} \propto v_{2} = \sqrt{\frac{T_{2}}{μ}}$
$∴ \frac{λ_{2}}{λ_{1}} = \sqrt{\frac{T_{2}}{T_{1}}} = \sqrt{\frac{(m_{1} + m_{2}) g}{m_{2} g}} = \sqrt{\frac{m_{1} + m_{2}}{m_{2}}}$

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